Height-scanning interferometry (HSI) employs broadband light sources to determine 3-D surface height profiles. HSI generates high-resolution profiles by combining two different pieces of information extracted from broadband interference patterns: coherence data and phase data. A low-resolution coherence height profile is derived from the localization of the interference effect, and a phase profile is calculated, e.g., from the interference phase itself, usually near the position of maximum interference signal strength (peak fringe contrast). By combining these two pieces of information, one can measure a high-resolution phase height profile of the surface without the fringe-order ambiguity normally associated with laser-based interferometry.
Fundamental to the success of high-resolution HSI is agreement between the data derived from coherence and phase information. Unfortunately, this is not always easily achieved. Optical distortions that vary with field position and object slope can deform the measured coherence and phase profiles in different ways, resulting in a mismatch that spoils prior-art techniques for determining fringe order in the phase profile. For example, a spherical object can contain erroneous changes in fringe order near the edges attributable in part to chromatic aberrations in the optics that distort the coherence profile. These and similar difficulties related to the mismatch between coherence and phase data can limit the range of application of high-resolution HSI.
In general, in one aspect, the invention features an analysis method for analyzing height-scanning interferometry data from a test surface. The method includes: calculating a coherence profile and a phase profile for the test surface based on the data; calculating an experimental phase gap map based on a difference between the phase profile and the coherence profile; filtering the experimental phase gap map to remove noise; and using the filtered phase gap map to determine a height profile of the test surface.
Embodiments of the method may include any of the following features.
The data may include an intensity signal I(xcex6,x) produced by interfering a measurement wavefront reflected from the test surface with a reference wavefront reflected from a reference surface, where the wavefronts are derived from a common source, xcex6 is a scan position for the reference surface, and x is a field position corresponding to an object position on the test surface. The coherence profile may be calculated from a localization of interference fringes in the intensity signal with respect to the scan position xcex6. Alternatively, the coherence profile may be calculated from a wavevector dependence of a phase xcfx86 of a transform (e.g., a Fourier transform) of I(xcex6,x) with respect to the scan position xcex6. The phase profile is may calculated from an interferometric phase of I(xcex6,x) at a nominal wavevector k0. For example, the phase profile may be calculated from a phase of a transform (e.g., a Fourier transform) of I(xcex6,x) with respect to the scan position xcex6 at a nominal wavevector k0.
The experimental phase gap map may be calculated by expressing the coherence profile and the phase profile in common units. For example, the coherence profile may be expressed in radians with respect to a nominal wavevector k0 according to "THgr"(x)=k0hC(x), where hC(x) is a surface height profile of the test surface derived from the coherence profile, and wherein the phase profile is calculated as the interferometric phase xcex8(x) in radians of the height scanning interferometry data at the nominal wavevector k0. In this case, the experimental phase gap map Gex(x) may be expressed as xcex8(x)xe2x88x92"THgr"(x). A difference between the experimental phase gap map Gex(x) and a theoretical phase gap map G(x)=xcex3(x)xe2x88x92k0xcfx84(x) can be indicative of agreement between the coherence profile of the test surface and the phase profile of the test surface, wherein xcex3(x) is a value of a phase offset at the nominal wavevector k0 produced by reflections from the test surface and elements of the interferometer used to measure the interferometry data, and xcfx84(x) is a value of linear dispersion in the phase offset with respect to wavevector. The method may further including determining values for xcex3(x) and xcfx84(x).
Calculating the experimental phase gap map may include smoothing the coherence profile to round edges in the coherence profile, and calculating the experimental phase gap map based on a difference between the phase profile and the smoothed coherence profile.
Filtering the experimental phase gap map may include calculating a global average of the experimental phase gap map. For example, calculating the global average may include calculating at least one trigonometric function (e.g., a sine and a cosine) for each of multiple points of the experimental phase gap map, averaging the results of each trigonometric function, and calculating an inverse trigonometric function based on each trigonometric average (e.g., arctan 2) to determine the global average of the experimental phase gap map.
In addition, filtering the experimental phase gap map may include calculating at least one trigonometric function for each of multiple points of the experimental phase gap map, smoothing the results of each trigonometric function over the multiple points, and calculating an inverse trigonometric function of the smoothed results to determine the filtered phase gap map. For example, calculating the at least one trigonometric function for the multiple points may include calculating a sine map and a cosine map based on the experimental phase gap map, and wherein the inverse trigonometric function is based on an arctan 2 function. Smoothing the results of each trigonometric functions may include using a convolution function or averaging the results among nearby points.
Furthermore, filtering the experimental phase gap map may includes smoothing the coherence profile to round edges in the coherence profile, and determining the filtered phase gap map based on a difference between the phase profile and the smoothed coherence profile.
Moreover, filtering the experimental phase gap may include a combinations of techniques, such as those described above. For example, filtering the experimental phase gap map may include calculating a variance map of the experimental phase gap, filtering the experimental phase gap map with each of multiple algorithms, and calculating the filtered phase gap map based on a locally weighted average of the algorithm outputs, wherein the local weights are based on the variance map. The variance map may include calculating at least one trigonometric function (e.g., sine and cosine) for each of multiple points of the experimental phase gap map, smoothing the results of each trigonometric function over the multiple points, and determining the variance map based on the smoothed trigonometric functions.
Using the filtered phase gap map may include connecting the filtered phase gap map to remove 2xcfx80 phase steps. Furthermore, using the filtered phase gap map may include fitting the connected filtered phase gap map to a polynomial function and using the polynomial function to improve an estimate for a height profile of the test surface.
Using the filtered phase gap map may further include determining a relative fringe order profile by determining a multiple of 2xcfx80 nearest to a difference between the experimental phase gap map and the connected filtered phase gap map. For example, using the filtered phase gap map further includes determining a relative height profile of the test surface based on the phase profile and the relative fringe order.
Moreover, using the filtered phase gap map may further include determining an absolute fringe order based on the experimental phase gap map, the connected filtered phase gap map, and a theoretical phase gap map G(x)=xcex3(x)xe2x88x92k0xcfx84(x), where the phase profile is calculated with respect to a nominal wavevector k0, xcex3(x) is a value of a phase offset at the nominal wavevector k0 produced by reflections from the test surface and elements of the interferometer used to measure the interferometry data, and xcfx84(x) is a value of linear dispersion in the phase offset with respect to wavevector. In this case, the method may further include determining values for xcex3(x) and xcfx84(x). Moreover, using the filtered phase gap map may further include determining an absolute height profile of the test surface based on the phase profile and the absolute fringe order.
In general, in another aspect, the invention features an interferometric analysis method including: providing a coherence profile and a phase profile derived from height-scanning interferometry data for a test surface; calculating a filtered phase gap map based on a difference between the phase profile and the coherence profile; and using the filtered phase gap map to determine a height profile of the test surface. Calculating the filtered phase gap map may include smoothing the coherence profile to round edges in the coherence profile, and calculating the filtered phase gap map based on a difference between the phase profile and the smoothed coherence profile. Furthermore, the method may include any of the features described further above in connection with the first inventive method.
In general, in another aspect, the invention features an interferometric analysis method for measuring surface roughness based on height-scanning interferometry data for a test surface. The method includes: calculating a coherence profile and a phase profile for the test surface based on the data; calculating an experimental phase gap map based on a difference between the phase profile and the coherence profile; and determining a surface roughness profile based on the experimental phase gap map. For example, determining the surface roughness profile may include calculating a variance map of the experimental phase gap map and determining the surface roughness profile based on the variance map. The variance map may include calculating at least one trigonometric function for each of multiple points of the experimental phase gap map, smoothing the results of each trigonometric function over the multiple points, and determining the variance map based on the smoothed trigonometric functions. Furthermore, the method may include any of the features described further above in connection with the first or second inventive methods.
In general, in another aspect, the invention features an interferometry system including: a height-scanning interferometer which during operation measures height-scanning interferometry data for a test surface; and an electronic processor coupled to the height-scanning interferometer. During operation the electronic processor: calculates a coherence profile and a phase profile for the test surface based on the data; calculates an experimental phase gap map based on a difference between the phase profile and the coherence profile; filters the experimental phase gap map to remove noise; and uses the filtered phase gap map to determine a height profile of the test surface.
In general, in another aspect, the invention features an interferometry system including: a height-scanning interferometer which during operation measures height-scanning interferometry data for a test surface; and an electronic processor coupled to the height-scanning interferometer. During operation the electronic processor: calculates a coherence profile and a phase profile for the test surface based on the data; calculates a filtered phase gap map based on a difference between the phase profile and the coherence profile; and uses the filtered phase gap map to determine a height profile of the test surface.
In general, in another aspect, the invention features an interferometry system including: a height-scanning interferometer which during operation measures height-scanning interferometry data for a test surface; and an electronic processor coupled to the height-scanning interferometer. During operation the electronic processor: calculates a coherence profile and a phase profile based on the data; calculates an experimental phase gap map based on a difference between the phase profile and the coherence profile; and determines a surface roughness profile based on the experimental phase gap map.
Any of these interferometry systems may further include features corresponding to any of the features described above in connection with the inventive methods.
Finally, in another aspect, the invention features a computer readable medium including a program that causes a processor to perform any of the inventive methods described above.
Embodiments of the invention may reduce sensitivity to a mismatch between coherence profile data and phase profile data. As a result, fringe orders can be more accurately determined and used to determine absolute height profiles. Additional embodiments of the invention may provide a measure of surface roughness based the phase gap analysis.
Other aspects, advantages, and features of the invention follow.